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3

$begingroup$

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following:

Let $(X,*)$ be a group and $(X,d)$ a metric space such that all the maps $ymapsto x*y$ are $d$-isometries (for example Euclidean space with translations), also let $S$ be a discrete state space. Now I define an infinitesimal rule on $X$ as a family of transition functions $Gamma_s_s>0$ with $Gamma_s:(B^d_s(e)to S)to S$ such that $$Gamma_s+t(f)=Gamma_s(xmapstoGamma_t(ymapsto f(x*y)))$$ for $s,t>0$. We could restrict the $Gamma_s$ to something like Borel measurable sets, but I think that's not necessary since we can always introduce some kind of error state $epsilonin S$ so that a cell falls into $epsilon$ if it sees a pattern it cannot handle (like Borel non-measurable).

My questions, does this make any sense, are there any non-trivial infinitesimal rules known on e.g. $mathbfR^2$?

continuity infinitesimals cellular-automata

share|cite|improve this question

edited 12 hours ago

fweth

asked Mar 28 at 19:21

fwethfweth

1,195713

$endgroup$

add a comment | 

3

$begingroup$

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following:

Let $(X,*)$ be a group and $(X,d)$ a metric space such that all the maps $ymapsto x*y$ are $d$-isometries (for example Euclidean space with translations), also let $S$ be a discrete state space. Now I define an infinitesimal rule on $X$ as a family of transition functions $Gamma_s_s>0$ with $Gamma_s:(B^d_s(e)to S)to S$ such that $$Gamma_s+t(f)=Gamma_s(xmapstoGamma_t(ymapsto f(x*y)))$$ for $s,t>0$. We could restrict the $Gamma_s$ to something like Borel measurable sets, but I think that's not necessary since we can always introduce some kind of error state $epsilonin S$ so that a cell falls into $epsilon$ if it sees a pattern it cannot handle (like Borel non-measurable).

My questions, does this make any sense, are there any non-trivial infinitesimal rules known on e.g. $mathbfR^2$?

continuity infinitesimals cellular-automata

share|cite|improve this question

edited 12 hours ago

fweth

asked Mar 28 at 19:21

fwethfweth

1,195713

$endgroup$

add a comment | 

$begingroup$

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following:

Let $(X,*)$ be a group and $(X,d)$ a metric space such that all the maps $ymapsto x*y$ are $d$-isometries (for example Euclidean space with translations), also let $S$ be a discrete state space. Now I define an infinitesimal rule on $X$ as a family of transition functions $Gamma_s_s>0$ with $Gamma_s:(B^d_s(e)to S)to S$ such that $$Gamma_s+t(f)=Gamma_s(xmapstoGamma_t(ymapsto f(x*y)))$$ for $s,t>0$. We could restrict the $Gamma_s$ to something like Borel measurable sets, but I think that's not necessary since we can always introduce some kind of error state $epsilonin S$ so that a cell falls into $epsilon$ if it sees a pattern it cannot handle (like Borel non-measurable).

My questions, does this make any sense, are there any non-trivial infinitesimal rules known on e.g. $mathbfR^2$?

continuity infinitesimals cellular-automata

share|cite|improve this question

edited 12 hours ago

fweth

asked Mar 28 at 19:21

fwethfweth

1,195713

$endgroup$

share|cite|improve this question

edited 12 hours ago

fweth

asked Mar 28 at 19:21

fwethfweth

1,195713

share|cite|improve this question

edited 12 hours ago

fweth

asked Mar 28 at 19:21

fwethfweth

1,195713

asked Mar 28 at 19:21

fwethfweth

1,195713

asked Mar 28 at 19:21

fwethfweth

1,195713